Ranz-Marshall relation

For a free-falling spherical particle of radius R moving with velocity u relative to a fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for species A) is DA, the Ranz-Marshall correlation relates the Sherwood number (Sh), which incorporates kAg, to the Schmidt number (Sc) and the Reynolds number (Re) ... 

For a reaction represented by A(g) + bB(s) —> produc1(g), derive the relation between time (t) of reaction and fraction of B converted (/B), if the particle is spherical with an initial radius R0, and the Ranz-Marshall correlation for kAg(R) is valid, where R is the radius at t. Other assumptions are given above. 

Droplet Heatup. A relation for the time required for droplet heatup, T h, can be derived based on the assumption that forced convection is the primary heat-transfer mechanism, and that the Ranz-Marshall equation for heat transfer to submerged spheres holds (34). The result is... 

In a gas—sohd CFB with heterogeneous reactions and mass transfer, in Hne with the structural characteristics of the SFM model (Hong et al, 2012), as shown in Fig. 12, the mass transfer and reaction in any local space can be divided into components of the dense cluster (denoted by subscript c), the dilute broth (denoted by subscript f), and in-between (denoted by subscript i), respectively. And these terms can be represented by Ri (1 = gc, gf, gi, sc, sf, si). Both the dense and dilute phases are assumed homogenous and continuous inside, and the dense phase is fiarther assumed suspended uniformly in the dilute phase in forms of clusters of particles. Then the mass transfer terms can be described with Ranz-Marshall-hke relations for uniform suspension of particles (Haider and Basu, 1988). In particular, the mesoscale interaction over the cluster will be treated as is for a big particle with hydrodynamic equivalent diameter of d. Due to dynamic nature of clusters, there are mass exchanges between the dilute and dense phases with rate ofTk (k = g, s), pointing outward from the dilute to the dense phase. 

This corresponds to the traditional Ranz and Marshall relation as... 

For forced convective heat transfer over a sphere in a uniform flow, a frequently used empirical relation was proposed by Ranz and Marshall (1952) as... 

Ranz and Marshall (13) carefully studied the evaporation of water droplets in still and moving air. They found that the steady-state temperature at the surface of a small evaporating drop was the wet bulb temperature of the surrounding air. Marshall considered the dependence of the effective mass transfer coefficient on air velocity to be determined largely by the boundary layer of less mobile air at the droplet surface, and he reported correlations which show that varied linearly with v1/2. Ranz and Marshall s data clearly show this relation, which we may write as Equation 5. 

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